Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

learn more… | top users | synonyms

3
votes
1answer
26 views

Split Factorial of n

How can I split integers up to n into two groups such that the difference of the product of each group is as low as possible? Is there a way to optimize the selection for each group in order to ensure ...
1
vote
2answers
37 views

What is the logic/theorem/derivation behind finding the exponent of p in n! By [n/p] + [n/p^2] + [n/p^3] + …? [duplicate]

The exponent of prime number of 3 in 100! is 48. It means 100! is divisible by $3^48$ $$E_3(100!) = \left\lfloor\frac{100}3\right\rfloor + \left\lfloor\frac{100}{3^2}\right\rfloor + ...
1
vote
2answers
60 views

Any shortcut to calculate factorial of a number (Without calculator or n to 1)?

I've been searching the internet for quite a while now to find anything useful that could help me to figure out how to calculate factorial of a certain number without using calculator but no luck ...
6
votes
3answers
411 views

Some trouble with the induction

Prove, that for any positive integer $n \geqslant 2$ we have the inequality $$ \frac{ 4^n }{ n+1 } < \frac{ (2n)! }{ (n!)^2 }.$$ For $n=2$ the inequality is true. Directly just take and ...
-1
votes
1answer
10 views

Common factors /greatest common factors /least common factor

See I am baffling with the concept of above mentioned title I saw one question they wrote find common factors /gcf/lcm of 42,294,882 Please help me out in getting the concept Thanks
0
votes
1answer
37 views

How do I prove the formula for multichoose?

In combinatorics, there is a formula "$n$ multichoose $k$", which is the way of making a multiset having $k$ elements choosing out of $n$ options. "$n$ multichoose $k$" is the same as "$(n+k-1)$ ...
0
votes
0answers
36 views

How to write $n!=a^{\alpha_0}(a+1)^{\alpha_1}(a+2)^{\alpha_2}\cdots(a+r)^{\alpha_r}$?

We know that $n!=n(n-1)(n-2)\cdots3\cdot2\cdot1, n\in \mathbb N$. Now I am willing to write $n!$ as $a^{\alpha_0}(a+1)^{\alpha_1}(a+2)^{\alpha_2}\cdots(a+r)^{\alpha_r}$ where $a, r, \alpha\in \mathbb ...
1
vote
1answer
49 views

Approximating $\frac{(kn)!}{(n!)^k}$

Is there any approximations for the form $$\frac{(kn)!}{(n!)^k},$$ where $n$ and $k$ are positive integers? $n$ is not necessary much larger than $k$?
0
votes
3answers
55 views

How do I derive $n!$ from this series?

I am reading a book where the following reduction is performed, but it's not explained exactly what is going on. I'm sorry if this is a dumb question, but I simply don't get how we are deriving the ...
8
votes
4answers
140 views

Is there an integral representation for $\frac{1}{n!}$?

I know that $n!$ has various integral representations, for instance the $\Gamma$ function. I was wondering if $\frac{1}{n!}$ has an integral representation.
1
vote
2answers
44 views

How do I evaluate this $ \lim \frac{\zeta(n)} {({n)!}} , n\to\infty $ if it was existed?

Is there someone who can show me how do I evaluate this limit $$ \lim_{n\to +\infty} \frac{\zeta(n)} {n!} $$ if it exists ? Thank you for any help.
6
votes
0answers
72 views

Could this approximation be made simpler ? Solve $n!=a^n 10^k$

I need to find the smallest value of $n$ such that $$\frac{a^n}{n!}\leq 10^{-k}$$ in which $a$ and $k$ are given (these can be large numbers). I set the problem as : solve for $n$ the equation ...
6
votes
1answer
88 views

Why is $!0 = 1$?

The subfactorial function is defined as: $$!n = n!\sum_{i=0}^n\dfrac {(-1)^i} {i!}$$ I was curious and wanted to find out what $!0$ came out to be. Since I couldn't use it in the sum above, I used a ...
0
votes
2answers
46 views

Simplifying expressions with factorials

If $$\large{a_n = \frac{x^n}{2^n n!}}$$ , Then find $$\large{ \frac{a_{n +1}}{a_n}}$$ .. I tried the following: $$\large{a_{n + 1} = \frac{x^n}{2^n n!} + \frac{2^n n!}{ 2^n n!} = \frac{x^n + 2^n ...
4
votes
3answers
103 views

How to find $\lim _{ n\to \infty } \frac { ({ n!) }^{ 1/n } }{ n } $? [duplicate]

How to find $\lim _{ n\to \infty } \frac { ({ n!) }^{ 1/n } }{ n } $ ? I tried taking using logarithm to bring the expression to sum form and then tried L Hospital's Rule.But its not working.Please ...
1
vote
2answers
60 views

How do I find the probability of some elements being together inside a randomly arranged set?

If I have a total of $n$ balls made of $k$ red balls and $(n-k)$ green balls and I arrange them all randomly in a line, how can I calculate the probability $x$ of a group of $y$ red balls being ...
2
votes
1answer
63 views

Why do the closest primes whose distance $d \gt 1$ to $c(n)=\frac{(n+1)!+n!}{2}$ have always $d \in \Bbb P$?

I have made the following observation: define the center of $n!$ and $(n+1)!$, $c(n)$, as the number located exactly in the middle of $(n+1)!$ and $n!$. Def: $\forall n \gt 2\ , \ ...
1
vote
1answer
33 views

Proving that $\Delta x^{(n)} = n x^{(n-1)}$

Define $\Delta f(x) = f(x+1) - f(x)$ (the difference operator). Define $x^{(n)} = x(x-1) \dots (x-n+1)$ (the falling factorial function). There's a rather simple theorem which shows that $\Delta ...
0
votes
1answer
39 views

What function does Wolfram Alpha plot instead of the factorial?

Look to the second graph where Wolfram Alpha gives a continuous factorial function: What is the second graph? It is not the gamma function, since that has $\Gamma(-1)=0!=1$.
-3
votes
1answer
86 views

Solve this equation : $(2x)! = (x)! (x+2)!$ [closed]

Solve this equation : $(2x)! = (x)! (x+2)!$
10
votes
6answers
244 views

Determine whether $\frac{1000!}{100!^{10}}$ is an integer

Can you give an idea, how to find out whether the result of ${1000!}/{100!^{10}}$ an integer. Modulo division? But what I met was about powers like $2^{100}/125$...
9
votes
2answers
70 views

When is a binomial coefficient a factorial, i.e. when is $\binom{m}{j} = n!$ for some $n,m,j$?

As stated in the title: when is a binomial coefficient a factorial, i.e. when is $\binom{m}{j} = n!$ for some $m,j,n$? I was thinking about this problem a couple of days ago because in all my years of ...
1
vote
1answer
52 views

Number of Divisors of N factorial

Say d(N) = Number of factors of N! Briefly: I wish to know if there is a Recurrence relation for this problem Now I wish to Know if there is a way to calculate d(N) in terms of previously calculated ...
6
votes
6answers
142 views

Prove $\lim\limits_{n\to\infty}\frac{1}{\sqrt[n]{n!}}=0$

I used $$(n!)^{\frac{1}{n}}=e^{\frac{1}{n}\ln(n!)}=e^{\lim\limits_{n\to\infty}\frac{1}{n}\ln(n!)}$$ Then using Stirling's approximation and L'Hospital's rule on ...
1
vote
1answer
80 views

Solving a little Diophantine equation:$(n-1)!+1=n^m$ [duplicate]

How can I solve this Diophantine equation: $$(n-1)!+1=n^m$$ with $n,m$ positive integers? From Wilson's theorem we can note that $n$ is a prime number. I proved to rewriting the equation ...
7
votes
2answers
142 views

Show that $\lim\limits_{n \to \infty} \frac{(n!)^{1/n}}{n}= \frac{1}{e}$ [duplicate]

Show that $$\lim_{n \to \infty} \left\{\frac{(n!)^{1/n}}{n}\right\} = \frac{1}{e}$$ What I did is to let $U_n = \dfrac{(n!)^{\frac{1}{n}}}{n}$ and $U_{n+1} = ...
0
votes
1answer
24 views

Find number of possible combinations from list of items [closed]

I have the following list: Product Icecream Banana Strawberry Vanilla The possible combinations (based on the spec) are 5: ...
2
votes
1answer
46 views

Total possible ways of representing n! as a sum of two or more consecutive positive integers.

I need to calculate total possible ways of representing $n!$ as a sum of two or more consecutive positive integers. Example : $3!=1*2*3=6$ and $6=1+2+3$ the only one possible way. Answer : $1$ The ...
0
votes
3answers
31 views

Permutations and Combinations equation [closed]

I cannot figure out how I am supposed to start in solving for $n$ in this equation. $$_nP_4=84(_nC_2)$$ Thank you!
2
votes
2answers
85 views

Total number of divisors of factorial of a number

I came across a problem of how to calculate total number of divisors of factorial of a number. I know that total number of divisor of a number $n= p_1^a p_2^b p_3^c $ is $(a+1)*(b+1)*(c+1)$ where ...
1
vote
2answers
41 views

Find all natural numbers $m,n$ which :$m!+n!+10$ is perfect cube?

I would be interest to invesitigate for all natural numbers $m,n$ which: $m!+n!+10$ is perfect cube ?
0
votes
0answers
32 views

estimations in the birthday paradox?

The birthday paradox is the famous following problem: What is the probability $p_n$ that at least $2$ persons amongst $n$ persons chosen at random have the same birthday? Leap years are not taken ...
1
vote
3answers
60 views

The derivative of $x!$ and its continuity

is the factorial of fractions and negative numbers defined? If yes, then what is its graph? Also please find its domain. Our teacher said the factorial of a fraction is the fraction itself. He also ...
6
votes
0answers
97 views

Partial sums of falling factorials

I want to know if there exists some way, approximate or exact, to do a partial sum of falling factorials of the kind: $$\sum_{k=i}^{n}(a+k)_{h}$$ where all are constants. And I'm interested too in ...
0
votes
0answers
45 views

Is there an easy way to calculate the elementary symmetric functions?

Hello I am interested in the question of what, generally, is the sum of the series of reciprocals of a series of numbers we know its sum. I have particular interest in the Zeta function, which I ...
0
votes
4answers
54 views

Prove that $n! \geq {\lfloor n/2 \rfloor}^{\lfloor n/2 \rfloor}$

I am trying to see why it holds that, for $n \in \mathbb{N},$ $$n! \geq {\lfloor n/2 \rfloor}^{\lfloor n/2 \rfloor}.$$ I would appreciate help to see this.
2
votes
3answers
61 views

Why is this equality involving factorials true?

$$ (n +1)! -1 +(n +1)(n +1)! = (n +2)! -1 $$ Can someone explain me how in the world is this true? :D Thanks (yes I'm trying to understand induction).
5
votes
1answer
115 views

Prove by combinatorial method that $ \frac{(2m)! \cdot (2n)!}{(m)! \cdot (n)! \cdot (m+n)!} $ is an integer [duplicate]

Prove that $$ \dfrac{(2m)! \cdot (2n)!}{(m)! \cdot (n)! \cdot (m+n)!} $$ is a positive integer, where $(m,n) \in \mathbb{Z^{+}}$ I have already solved it using Legendre's Formula ...
2
votes
4answers
107 views

How many zeros are there in the number $50!$? [duplicate]

How many zeros are there in the number $50!$? My attempt: The zeros in every number come from the 10s that make up the number. The 10s are, in turn, made up of 2s and 5s. So: $\frac{50}{5*2} = 5$ ...
0
votes
1answer
35 views

Maximizing the following function

I need to find values of $k_1$, $k_2$ and $k_3$ that maximize $C^{m_1}_{mm_1} \cdot C^{m_2}_{k_1-mm_1} \cdot C^{n_1}_{nn_1} \cdot C^{n_2}_{k_2-nn_1} \cdot C^{p_1}_{pp_1} \cdot C^{p_2}_{k_3-pp_1}$ ...
6
votes
2answers
144 views

Proof of an identity of $n!$

I came up (numerically) with an identity concerning n! and I was wondering about a proof of it. Here it is: \begin{align} \ n! &= \sum_{r=0}^{n} { \binom{n}{r} (-1)^r(k-r)^n } \quad \forall n ...
0
votes
0answers
42 views

Modular arithmetic, specifically n! mod m

Is there a theorem that makes solving $$ n! \equiv x \mod m $$ knowing that both $n$ and $m$ are prime? And if not, what would be the best way to go about finding $x$? cheers
1
vote
3answers
84 views

Trying to show that $\sum_{1}^{\infty} \frac{n^n}{n!} $ diverges.

I have been trying to prove this using the ratio test $|\frac{A_{n+1}}{A_n}|$ , which leads me to this expression: $$\left|\frac{(n+1)^{n+1}}{(n+1)!}\cdot ...
1
vote
3answers
71 views

Calculate $\lim_{n\to\infty}\frac{5n^n}{3n!+3^n}$

Could I please have a hint for finding the following limit?$$\lim_{n\to\infty}\frac{5n^n}{3n!+3^n}$$
1
vote
5answers
108 views

Evaluate $ \lim_{n\to\infty} \frac{(n!)^2}{(2n)!} $ [duplicate]

I'm completely stuck evaluating $ \lim_{n\to\infty} \frac{(n!)^2}{(2n)!} $ how would I go about solving this?
0
votes
2answers
48 views

Trying to determine the number of possible combinations for a password

OVERVIEW: Making a secure password. People tend to use dictionary words as a basis for their passwords. People tend to make minor substitutions on their passwords (password -> p@$$w0rd) Assuming ...
2
votes
5answers
94 views

To evaluate $\lim_{n\to\infty} \dfrac{10^n}{n!} .$

I am having a lot of trouble evaluating $$\lim_{n\to\infty} \dfrac{10^n}{n!} .$$ I know intuitively that $n!$ is much larger and this expression should go to zero but I just can't figure out how to ...
1
vote
2answers
126 views

What is the triple factorial of a negative number, e.g., $-2$?

The triple factorial of a positive integer is computed as $7!!! = 7\cdot 4\cdot 1$. I'm interested in the value of $$(-2)!!!$$ I tried to find this value by using the Wolfram, but I found the ...
4
votes
4answers
161 views

Simplify the expression (combination and factorial)

Simplify the following expression: $\binom{n+1}{3} * \frac{(n-1)! + (n-2)!}{(n+1)!}$ My attempt: $\binom{n+1}{3} * \frac{(n-1)! + (n-2)!}{(n+1)!} = \frac{(n+1)!}{3!(n+1-3)!} * \frac{(n-1)! + ...
0
votes
1answer
56 views

number of ways to choose pairs of nonadjacent people from $2k$ people sitting in a circle

The following is problem 19 in Chapter 2 from Richard Stanley's Enumerative Combinatorics, vol. 1 (2nd ed.): Suppose that $2k$ persons are sitting in a circle. In how many ways can they form pairs if ...